JERUSALEM/BEIRUT, July 24 (Reuters) - Israel said it shot down a Syrian warplane that crossed into the occupied Golan Heights on Tuesday, but Damascus said the jet was fired on as it took part in sorties against rebels within Syria.

The incident added new fuel to weeks of tensions over the Golan, a strategic plateau between the two old enemies and where Israel has been on high alert as Syrian government forces, supported by Russia, close in to regain rebel-held ground.

Israel worries Syrian President Bashar al-Assad may defy a 1974 U.N. armistice that demilitarised much of the Golan, or let his Iranian and Lebanese Hezbollah reinforcements deploy there.

Israeli Prime Minister Benjamin Netanyahu said the Syrian jet had been in "gross violation of the separation of forces between us an Syria". Israel "took appropriate action" he said.

For the second time in as many days, Israeli sirens sounded on the Golan and witnesses saw the contrails of two missiles flying skyward. The military said it fired Patriot interceptor missiles at a Syrian Sukhoi jet that crossed 2 km (1 mile) into Israeli-controlled air space, after first trying to warn it off.

U.N. peacekeepers "observed burning debris falling from an aircraft" some 10 km (6 miles) inside Syrian territory southeast of the Golan buffer zone, their agency said in a statement.

The warplane's pilot was killed, a non-Syrian source close to the Syrian government told Reuters.

The source said the plane fell in a pocket of territory held by Islamic State-affiliated forces adjoining the occupied Golan Heights. The Syrian army is advancing into this pocket to try to bring the whole of southwest Syria back under its control.

Syrian state media said the plane was targeted by Israel and hit while conducting raids in Syrian-controlled air space.

"The Israeli enemy confirms its support for the armed terrorist groups," the official news agency SANA quoted a military source as saying. Lebanon's pro-Damascus TV station al-Mayadeen said the plane fell 15 km (8 miles) inside Syrian territory and did not enter the air space of the occupied Golan.

The spiralling Israeli-Syrian tensions have spurred intercession by Moscow, which sent Foreign Minister Sergei Lavrov and its top general on Monday for talks with Netanyahu.

Nickolay Mladenov, a U.N. envoy for the Middle East, warned on Tuesday of "a disturbing trajectory of increasingly frequent and dangerous confrontations" between Israel and Syria.

"I call on all parties to abide by all the provisions of the 1974 agreement," he told the U.N. Security Council.

Before Tuesday's shoot-down there was an escalation of the Syrian fighting close to the Golan, Israeli military spokesman Lieutenant-Colonel Jonathan Conricus said.

"We issued warnings and messages through different channels in various languages numerous times over the day in order to avoid any miscalculations, misunderstandings or any violations of Israeli air space," he told reporters.

On Monday, Reuters witnesses on the southern edge of the Israeli-occupied Golan spotted numerous warplanes and helicopters in the skies over Syrian territory, some of them dropping bombs. Anti-aircraft return fire could also be seen.

In February, an Israeli F-16 jet was brought down by Syrian anti-aircraft fire.

That warplane crashed in northern Israel while returning from a bombing raid on what Israel said was an Iranian military installation in Syria. Both pilots ejected, injured. (Additional reporting by Stephen Farrell in Jerusalem, Laila Bassam in Beirut and Michelle Nichols in the United Nations; Writing by Dan Williams; editing by William Maclean/Mark Heinrich)